New York City Mayor Bill de Blasio is due to give his second State of the City speech on Tuesday after a year in which his fraught relationship with police has often overshadowed his campaign promises to reduce economic and social inequality.

Before the speech at a Manhattan college, de Blasio's office released short videos online of the mayor drafting his remarks in his office with his advisers, describing the points he wants to make.

"Affordable housing will grow, pre-K will grow, police-community relations changes will grow," he says in the video, turning a baseball in his hands as aides type on their laptop computers and shuffle papers.

De Blasio launched an ambitious expansion of pre-kindergarten, or "pre-K", for more than 50,000 young children last year, fulfilling one of his main campaign pledges.

But his greatest immediate challenge has come with his promise to mend frayed relations between police and black and Latino New Yorkers, who were stopped and frisked by the police in disproportionate numbers under the previous administration.

After an unarmed black, Eric Garner, was killed by police who put him in a banned chokehold on a Staten Island sidewalk in July, police union leaders railed at the mayor, saying he was too supportive of the force's critics.

The rift between City Hall and rank-and-file officers only deepened in December after a man, described by authorities as unstable and suicidal, shot two New York police officers to death in an ambush. The man, who killed himself soon after, had written online that he wanted to avenge Garner's death.

Many police officers took to turning their backs on the mayor at public events, including at the slain officers' funerals. Arrest numbers plummeted for a few weeks beginning in December in a work slowdown, although policing activity has since begun to return to normal levels. (Reporting by Jonathan Allen; Editing by Doina Chiacu)

New York City Mayor Bill de Blasio is due to give his second State of the City speech on Tuesday after a year in which his fraught relationship with police has often overshadowed his campaign promises to reduce economic and social inequality.Before the speech at a Manhattan...